Game Theory and Me

“Game Theory”!? If you know what this is about, you are not alone. If you don’t, you are not alone either. When I tell people that my research is about game theory application in road-bridge transportation network. Reaction from most people are that their facial expression tells me they lost interest about my subject, or they still look at me, but I don’t have any response from them. Sometimes I get frustrated when people lose interest for my topic.  It is not a fun or hot topic like Artificial intelligence(AI) or Game of Thrones. I want to write something of game theory not about my research, but about how to apply game theory to our personal life.

In this field, eleven game-theorists have won the economics Nobel Prize. John Maynard Smith was awarded the Crafoord Prize for his application of game theory to biology. John Nash sets a milestone in the development of game theory.  His major contribution is the definition and properties of Nash equilibrium, which is a crucial concept of a non-cooperative game (and I will discuss below).

“A Beautiful Mind,” is a film based on John Nash life. This is a scene explaining game theory in which Russell Crowe, playing John Nash, is at a bar with three friends. A beautiful blond woman  walks in with four brunette friends. John and his friends are attracted by the blond,  and  his friends joke about which of them would successfully charm the blonde. Surprisely  John Nash concludes they should do the opposite: Ignore her. “If we all go for the blonde,” he says, “we block each other and not a single one of us is going to get her. So we go for her friends, but they will all give us the cold shoulder because nobody likes to be second choice. But what if no one goes to the blonde? We don’t get in each other’s way and we don’t insult the other girls. That’s the only way we win.” This episode gives us example how game theory exits in daily life.

Prisoner’s Dilemma

A simpler example is what is known as the Prisoner’s Dilemma. Two members of a criminal gang are arrested and offered a deal: “If you confess and testify against your accomplice, we’ll let you go and throw the book at the other guy — 20 years in prison.” If both remain quiet, the prosecutors cannot prove the more serious charges and both would spend just a year behind bars for lesser crimes. If both confess, the prosecutors would not need their testimony, and both would get five-year prison sentences.

At first glance, remaining silent might seem the best option. If both did so, both would get off fairly lightly. But as a rational person, which is a basic assumption of game theory, confess is best choice regardless what your accomplice chooses.

This type of problem is called a non-cooperative game, which means the two prisoners cannot communicate with each other. Without knowing what the other person is doing, each is faced with this choice: If he confesses, he could end up with freedom or five years in prison. If he stays quiet, he goes to prison for one year or 20 years.  Prisoner’s dilemma is a widely used example in most textbook of game theory to illustrate the calculation of Nash equilibrium. We will use this example to introduce some concepts of game theory.  In the Prisoner’s dilemma game, two members of a criminal gang are players. Two players game are common, but a game can have more than two players.

Another important concept is payoff. In a game, payoffs are numbers which represent the motivations of players. Payoffs can be in any quantifiable form, from dollar to “utility”. Confess and remain silence are strategies set. In game theory, player’s strategy is any of the options the player can choose in a setting where the outcome depends not only on his own actions but on the actions of others.

Ok, you probably have the question for a while: What is Nash equilibrium? It is the optimal outcome of a game where no player gains anything by unilaterally changing his own strategy, then current set of strategy and corresponding payoff forms a Nash equilibrium. In the Prisoner’s dilemma game, (confess,confess) is the Nash equilibrium, since it is the set of strategies that maximize each player’s payoff given the other prison’s strategy. Also confess is dominant strategy of the game. Rule of thumb: never play a dominated strategy in a game!

A little bit of Game Theory in my life… (to the hymn of Mambo Number 5)

Now we know the basic of game theory. What we can do for our personal life by using it? By wooing a girl in a bar? Or by getting less sentence when we get arrested? These two episodes likely for most people are never-happen-in-real-life. It is not easy to recognize strategic situation and change no-win situation to mutually beneficial outcomes. And you can’t  be a master of game theory with one-day’s lesson, but you don’t need a Ph.D to apply basic game theory in everyday life.

A political scientist and professor at NYU, Bruce Bueno de Meaquita, experimented to purchase car by applying game theory.

Here is his strategy: First, you establish a radius for how far you are willing to travel to buy a car. Then you call a dealer to ask best price. The dealer most of time tell you: You can’t buy a car over the phone.  You should elaborate the conversation to this way: “I know I can buy a car this way, because I know that many cars have been purchased this way. If you do not quote a price for me, I understand that you are telling me that you know you don’t have the best price, I appreciate you saving my time.”

Then you can explain that you will buy a car from whoever gives you the lowest price and you will show up with a check in hand without discussing the price.  A lot of people react by this way: “No way, it is not going to work.” In Bruce’s video, he said he has bought 11 cars using this strategy! One of his students and Irish reporter have uses his strategy and saved thousands on their purchase.

Let’s give a quick analysis for Bruce’s strategy. In this car-purchasing game, dealers you called on the phone are players. To simplify the problem, we assume that there are only two dealers compete in this game. Each leader has two strategies: a higher price or a lower price. If you buy car from dealer offering lower price, another dealer payoff is zero. In the bi-matrix, the first value of inside parenthesis is the payoff of dealer 1 and the second is payoff of dealer 2.  Apparently, if a dealer offers higher price, his incentive is zero. Low price is dominant strategy.

Of course, the reality of car purchasing involves many other factors, but Game theory demonstrates that getting multiple dealer involved is one of the most effective strategies for a car buyer!

If you end up using this approach to help yourself buy a car, please let all of us at the Analytics Lab @ OU know about it!